Mixed Convection Peristaltic Flow of Third Order Nanofluid with an Induced Magnetic Field

This research is concerned with the peristaltic flow of third order nanofluid in an asymmetric channel. The governing equations of third order nanofluid are modelled in wave frame of reference. Effect of induced magnetic field is considered. Long wavelength and low Reynolds number situation is tackled. Numerical solutions of the governing problem are computed and analyzed. The effects of Brownian motion and thermophoretic diffusion of nano particles are particularly emphasized. Physical quantities such as velocity, pressure rise, temperature, induced magnetic field and concentration distributions are discussed.


Introduction
Peristaltic motion is now an important research topic due to its immense applications in engineering and physiology. This type of rhythmic contraction is the basis of peristaltic pumps that move fluids through tubes without direct contact with pump components. This is a particular advantage in biological/medical applications where the pumped material need not to contact any surface except the interior of the tube. The word ''peristalsis'' comes from a Greek word ''Peristaltikos''which means clasping and compressing. The peristaltic flow has specific involvement in the transport of urine from kidney to the bladder, chyme movement in gastrointestinal tract, movement of ovum in the female fallopian tubes, blood circulation in the small blood vessels, roller and finger pumps, sanitary fluid transport and many others. Latham [1] and Shapiro et al. [2] reported initial studies for the peristaltic flow of viscous fluid. Since then ample attempts have been made for peristalsis in symmetric flow geometry (seefewrecentstudies [3][4][5][6][7][8]). Recently, physiologists argued that the intra-uterine fluid flow (because of mymometrical contractions) represents peristaltic mechanism and the myometrical contractions may appear in both asymmetric and symmetric channels [9]. Hence some researchers [10][11][12][13][14][15] discussed the peristaltic transport in an asymmetric channel with regard to an application of intrauterine fluid flow in a nonpregnant uterus.
Heat transfer in cooling processes is quite popular area of industrial research. Conventional methods for increasing cooling rates include the extended surfaces such as fins and enhancing flow rates. These conventional methods have their own limitations such as undesirable increase in the thermal management system's size and increasing pumping power respectively. The thermal conductivity characteristics of ordinary heat transfer fluids like oil, water and ethylene glycol mixture are not adequate to meet today's requirements. The thermal conductivity of these fluids have key role in heat transfer coefficient between the heat transfer medium and heat transfer surface. Hence many techniques have been proposed for improvement in thermal conductivity of ordinary fluids by suspending nano particles in liquids. The term ''nano'' introduced by Choi [16] describes a liquid suspension containing ultra-fine particles (diameter less than 50 nm). The nanoparticle can be made of metal, metal oxides, carbide, nitride and even immiscible nano scale liquid droplets. Although the literature on flow of viscous nanofluid has grown during the last few years (see [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] andmanyrefs.therein) but the information regarding peristaltic flow of nano fluids is yet scant. To our information, Akbar and Nadeem [33] studied the peristaltic flow of viscous nanofluid with an endoscope. Influence of partial slip in peristaltic flow of viscous fluid is explained by Akbar et al. [34].
The aim of present study is to venture further in the regime of peristalsis for fluids with nanoparticles. Therefore we examine here the mixed convective peristaltic transport of third order nanofluid in an asymmetric channel. Channel asymmetry is produced by peristaltic waves of different amplitude and phases. Mathematical modelling involves the consideration of induced magnetic field, Brownian motion and thermophorsis effects. Numerical solution of nonlinear problem is obtained using shooting method. Limiting case for viscous nanofluid in symmetric channel is also analyzed. Detailed analysis for the quantities of interest is seen.

Physical Model
Extra stress tensor S S for third order fluid model is given by in which I, p, m, S and A 1 , A 2 respectively stand for the identity tensor, the pressure, the fluid dynamic viscosity, the extra stress tensor and the first and second Rivlin Ericksin tensors in which the In the absence of displacement current, the Maxwell's equations are + : E~0, +|E~{m e LH Lt , +|H~J, + : H~0, ð4Þ  J~s Ezm e V|H ð Þ ð Þ : ð5Þ

Mathematical Formulation
Consider third order nanofluid in an asymmetric channel of width d 1 zd 2 . Let c be the speed by which sinusoidal wavetrains propagate along the channel walls. The X X and Y Y -axes in the rectangular coordinates X X , Y Y ð Þ system are taken parallel and transverse to the direction of wave propagation, respectively. A constant magnetic field of strength H 0 acts in the transverse direction resulting in an induced magnetic field Further the lower wall is maintained at temperature T 0 and nano particles concentration C 0 while the temperature and nanoparticles concentration at the upper wall are T 1 and C 1 respectively. The wall surfaces satisfy   is subjected to the symmetric channel with waves out of phase and the waves are in phase for w~p: Further l is the wavelength, t t the time and a a 1, a a 2 , d d 1 , d d 2 and w satisfy a a 2 1 z a a 2 2 z2 a a 1 a a 2 cos w Denoting the velocity components U U and V V along the X X and Y Y {directions in the fixed frame, one can write V as The fundamental equations governing the flow of an incompressible fluid are  Peristaltic Flow of Third Order Nano Fluid in which r denotes the density of fluid, D T the thermophoretic diffusion coefficient, T the temperature, C C the concentration, ' the thermal conductivity, g the acceleration due to gravity, p p is the pressure, D B is Brownian diffusion coefficient, t~(rc 1 ) p =(rc 1 ) f the ratio of the specific heat capacity of the nanoparticle material and heat capacity of the fluid, k the thermal diffusivity, c 1 is the volumetric volume expansion coefficient, and r p is the density of the particle, Y Y are components of extra stress tensor S and z~sm e is the magnetic diffusivity.
To facilitate the analysis, we introduce the following transformations between fixed and wave frames x x~ X X {c t t, y y~ Y Y , p p( x x, y y)~ P P( in which ( u u, v v) are the velocity components in the wave frame. Equations (1)-(12) in terms of above transformations give Defining Gc,Pr,Gt,Nb,Nt,R m , M as mass Grashof number, Prandtl number, local temperature Grashof number, Brownian   Nb~t and then employing long wavelength and low Reynolds number approximation, the dimensionless forms of above equations in terms of stream function Y and magnetic force function w can be expressed as Lp Lx~L where C~f 1 zf 2 : The dimensionless boundary conditions are given by with a 2 zb 2 z2ab cos w 0 ƒ 1zd ð Þ 2 . The dimensionless time mean flow rate F in the wave frame is related to the dimensionless time

Pumping characteristics
This subsection illustrates the behavior of emerging parameters Nt, Nb, C,Gt, Gc, and M on pressure rise per wavelength DP l . The dimensionless pressure rise per wavelength versus timeaveraged flux h has been plotted in the Figs. 1-6. Here the upper right-hand quadrant (I) denotes the region of peristalsis pumping, where hw0 (positive pumping) and DP l w0 (adverse pressure gradient). Quadrant (II), where DP l v0 (favorable pressure gradient) and hw0 (positive pumping), is designated as augmented flow (copumping region). Quadrant (IV ), such that DP l w0 (adverse pressure gradient) and hv0, is called retrograde or backward pumping. The flow is opposite to the direction of the  There is an inverse linear relation between DP l and h. It is noticed from Figs. 1-2 and 4-5 that Dp l increases with Nt, Nb,Gt and Gc in all the pumping regions. Fig. 3 shows that pumping rate increases by increasing C in pumping region. There are specific values of h for which there is no difference between viscous and third order nanofluids. On the other hand, in the copumping region the pumping rate decreases with the increase in Deborah number. Fig. 6 shows that Dp l decreases with M in copumping region.

Flow characteristics
The variations of C, Gc, Gt,h, Nt, Nb, and Pr on the velocity have been plotted in this subsection. Fig. 7 shows that there is an increase in velocity at the centre of the channel when C increases. We see a little influence of Deborah number on velocity near the  walls of channel. However, magnitude of the velocity of third order nanofluid is more than viscous nanofluid. Figs. 8 and 9 depict the influence of local and mass Grashof number. Clearly the velocity increases near the lower wall. Increase in h supports the motion in the channel which is shown in Fig. 10. Fig. 11 shows the influence of Nt on velocity distribution. Interestingly an increasing thermophoresis leads to an increase in the fluid velocity at the lower wall of channel. There is a considerable variation near the walls y~h 1 and y~h 2 for Nb and Pr (Figs. 12-13).

Heat transfer characteristics
Effect of heat transfer on peristalsis is shown in the Figs. 14-16. Figs. 14 and 15 depict the effects of Brownian motion parameter (Nb) and thermophoresis parameter (Nt) on the temperature profile. One can observe that the temperature profile is an increasing function of Nb and Nt between the walls y~h 1 and y~h 2 . In Fig. 16

Mass transfer characteristics
Influence of mass transfer on peristalsis is shown in the Figs. 17-19. Figs. 17 and 18 depict that the concentration distribution increases at the upper and lower walls of channel when Nt and Nb are increased. Fig. 19 shows the effect of Pr on the concentration when the other parameters are fixed. It shows increasing behavior of Pr on concentration distribution V near the walls h 1 and h 2 :

Induced magnetic field characteristics
The variations of C, R m and M on the induced magnetic field have been plotted in the Figs. 20-22. Fig. 20 shows that there is an increase in h x when C increases. We see that magnitude of the induced magnetic field in third order nanofluid is more than

Trapping
Trapping phenomenon is shown in the Figs. 23 and 24 for different values of Gt and Nt respectively. Trapping is an interesting aspect of peristaltic motion. It is the formation of a bolus of fluid by the closed streamlines. Fig. 23 is made for increasing values of Gt: We note that trapping exists for Gt~0,0:1, 0:2 in the upper part of channel. It is observed that number of closed streamlines circulating the bolus reduce in number as we increase the values of local Grashof number. Meanwhile size of trapped bolus increases. Streamlines are plotted in Fig. 24 to see the effects of thermophoresis parameter (Nt): Clearly, the size of trapped bolus increases when Nt increases from 0 to 11: An upper shift and flatness of bolus along with reduced closed streamlines is observed.

Conclusions
A detailed analysis is presented for peristaltic transport of third order nanofluid in an asymmetric channel with an induced magnetic field and mixed convection. The main findings of the presented study are listed below.